Trigonometric Identities, Equations, and Applications: Study Guide for Exam 3

Trigonometric Identities and Proofs

Fundamental Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables for which both sides are defined. Mastery of these identities is essential for simplifying expressions and solving equations in trigonometry.

• Pythagorean Identities: These relate the squares of sine, cosine, and tangent functions.

  • $\sin^2\theta + \cos^2\theta = 1$

  • $1 + \tan^2\theta = \sec^2\theta$

  • $1 + \cot^2\theta = \csc^2\theta$

• Reciprocal Identities:

  • $\sin\theta = \frac{1}{\csc\theta}$

  • $\cos\theta = \frac{1}{\sec\theta}$

  • $\tan\theta = \frac{1}{\cot\theta}$

• Quotient Identities:

  • $\tan\theta = \frac{\sin\theta}{\cos\theta}$

  • $\cot\theta = \frac{\cos\theta}{\sin\theta}$

Proving Trigonometric Identities

To prove a trigonometric identity, manipulate one side of the equation using known identities until it matches the other side. For example:

• Example: Prove $\cos^2x + \sin^2x = 1$.

  • This is the fundamental Pythagorean identity, derived from the unit circle definition.

• Example: Prove $\cos^2x = 1 - \sin^2x$.

  • Rearrange the Pythagorean identity: $\cos^2x = 1 - \sin^2x$.

Sum and Difference Formulas

Sum and Difference Identities

These formulas allow you to express the sine, cosine, and tangent of sums or differences of angles in terms of the sines and cosines of the individual angles.

• Sine:

  • $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

• Cosine:

  • $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

• Tangent:

  • $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

These identities are useful for evaluating trigonometric functions at non-standard angles and for simplifying expressions.

Examples and Applications

• Example: Find $\sin(75^\circ)$.

  • Express as $\sin(45^\circ + 30^\circ)$ and apply the sum formula.

  • $\sin(75^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ)$

Double-Angle and Half-Angle Formulas

Double-Angle Formulas

Double-angle identities express trigonometric functions of $2\theta$ in terms of $\theta$.

• $\sin(2\theta) = 2\sin\theta\cos\theta$

• $\cos(2\theta) = \cos^2\theta - \sin^2\theta$

• Alternate forms:

  • $\cos(2\theta) = 2\cos^2\theta - 1$

  • $\cos(2\theta) = 1 - 2\sin^2\theta$

• $\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$

Half-Angle Formulas

Half-angle identities allow you to find the sine, cosine, or tangent of $\frac{\theta}{2}$.

• $\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$

• $\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$

• $\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$ or $\frac{\sin\theta}{1 + \cos\theta}$

The sign depends on the quadrant in which $\frac{\theta}{2}$ lies.

Trigonometric Equations and Exact Values

Solving Trigonometric Equations

To solve trigonometric equations, use algebraic manipulation and identities to isolate the variable. Consider the domain and possible multiple solutions due to periodicity.

• Example: Solve $\sin x = \frac{1}{2}$ for $x$ in $[0, 2\pi]$.

  • Solutions: $x = \frac{\pi}{6}, \frac{5\pi}{6}$

Finding Exact Values Using Identities

Use sum, difference, double-angle, or half-angle identities to find exact values for trigonometric functions at specific angles.

• Example: Find $\sin(15^\circ)$ using half-angle identity.

  • $\sin(15^\circ) = \sin\left(\frac{30^\circ}{2}\right) = \pm\sqrt{\frac{1 - \cos 30^\circ}{2}}$

Using Reference Angles and Quadrants

Reference Angles and Signs

The sign of a trigonometric function depends on the quadrant in which the angle lies. Reference angles help determine the value and sign of the function.

• Quadrant I: All functions positive

• Quadrant II: Sine positive, cosine and tangent negative

• Quadrant III: Tangent positive, sine and cosine negative

• Quadrant IV: Cosine positive, sine and tangent negative

Application: Evaluating Expressions and Proofs

Examples of Application

• Given: $\sin\theta = \frac{3}{5}$, $\theta$ in Quadrant II. Find $\cos\theta$ and $\tan\theta$.

  • $\cos\theta = -\sqrt{1 - \sin^2\theta} = -\sqrt{1 - (\frac{3}{5})^2} = -\frac{4}{5}$

  • $\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{3/5}{-4/5} = -\frac{3}{4}$

Table: Common Trigonometric Values

The following table summarizes the exact values of sine, cosine, and tangent for commonly used angles.

Angle (Degrees)	Angle (Radians)	$\sin\theta$	$\cos\theta$	$\tan\theta$
0	0	0	1	0
30	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
45	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	1
60	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90	$\frac{\pi}{2}$	1	0	Undefined

Summary of Key Strategies

• Memorize fundamental identities and common values.

• Practice proving identities by manipulating one side to match the other.

• Apply sum, difference, double-angle, and half-angle formulas to simplify and evaluate expressions.

• Use reference angles and quadrant information to determine signs and values.

• Check solutions for trigonometric equations within the specified domain.